Instability of degenerate solitons for nonlinear
Schr¨
odinger equations with derivative
By
Noriyoshi FUKAYA and Masayuki HAYASHI
February 2021
R
ESEARCH
I
NSTITUTE FOR
M
ATHEMATICAL
S
CIENCES
Schr¨
odinger equations with derivative
Noriyoshi Fukaya and Masayuki Hayashi
Abstract. We consider the following nonlinear Schr¨odinger equation with de-rivative:
(1) iut= −uxx− i|u|2ux− b|u|4u, (t, x) ∈ R × R, b ∈ R.
If b = 0, this equation is a gauge equivalent form of the well-known derivative nonlinear Schr¨odinger (DNLS) equation. The equation (1) for b ≥ 0 has de-generate solitons whose momentum and energy are zero, and if b = 0, they are algebraic solitons. Inspired from the works [29, 8] on instability theory of the L2-critical generalized KdV equation, we study the instability of degenerate soli-tons of (1) in a qualitative way, and when b > 0, we obtain a large set of initial data yielding the instability. The arguments except one step in our proof work for the case b = 0 in exactly the same way, and in particular the unstable di-rections of algebraic solitons are detected. This is a step towards understanding the dynamics around algebraic solitons of the DNLS equation.
Contents
1. Introduction 1
2. Structure of the linearized operator 7
3. Modulation theory 12
4. Virial identities 19
5. Proof of instability 21
Appendix A. Relation to instability theory on (gKdV) 22
Acknowledgments 23
References 23
1. Introduction
We consider the following nonlinear Schr¨odinger equation with derivative: iut= −uxx− i|u|2ux− b|u|4u, (t, x) ∈ R × R,
(1.1)
where b ∈ R, and u is the complex-valued unknown function of (t, x) ∈ R × R. It is well-known (see [39]) that (1.1) is locally well-posed in the energy space H1(R) and the following three quantities
E(u) := 1 2kuxk 2 L2 − 1 4(i|u| 2u x, u)L2− b 6kuk 6 L6, (Energy) M (u) := kuk2L2, (Mass) P (u) := (iux, u)L2, (Momentum) Date: February 26, 2021. 1
are conserved by the flow. Here the inner product (·, ·)L2 is defined by
(v, w)L2 = Re
Z
R
v(x)w(x) dx,
and we regard L2(R) as a real Hilbert space. The equation (1.1) is L2-critical (mass-critical) in the sense that (1.1) is invariant under the scaling
uλ(t, x) = λ1/2u(λ2t, λx),
which satisfies kuλ(0)kL2 = ku(0)kL2. By using the energy functional, (1.1) is
rewritten as
(1.2) iut(t) = E0(u(t)).
When b = 0 the equation (1.1) is sometimes referred to as the Chen-Lee-Liu equa-tion [5]. This equaequa-tion is a gauge equivalent form of the well-known derivative nonlinear Schr¨odinger equation
iψt= −ψxx− i(|ψ|2ψ)x, (t, x) ∈ R × R,
(DNLS)
which was introduced as a model in plasma physics [32, 33] and shown to be com-pletely integrable [21]. The equation (1.1) can be considered as a generalization of (DNLS) while preserving L2-criticality and the Hamiltonian structure (1.2).
The equation (1.1) admits a two-parameter family of solitons1 uω,c(t, x) = eiωtφω,c(x − ct), where (ω, c) ∈ R2 satisfies (−2√ω < c ≤ 2√ω if b > −3/16, −2√ω < c < −2κ∗ √ ω if b ≤ −3/16, (1.3) κ∗= κ∗(b) := r −γ 1 − γ = r 3 + 16b 16b ∈ (0, 1) when b ≤ −3/16, γ = γ(b) := 1 + 16 3 b, and φω,c is explicitly written as
φω,c(x) = Φω,c(x) exp ic 2x − i 4 Z x −∞ Φω,c(y)2dy , Φω,c(x) = 2(4ω − c2) pc2+ γ(4ω − c2) cosh(√4ω − c2x) − c !1/2 if −2√ω < c < 2√ω, 4c (cx)2+ γ 1/2 if c = 2√ω. We note that φω,c∈ H1(R) is the nontrivial solution of the stationary equation
(1.4) −φ00+ ωφ + ciφ0− i|φ|2φ0− b|φ|4φ = 0, x ∈ R, and that Φω,c is the positive even solution of
(1.5) −Φ00+ ω − c 2 4 Φ + c 2|Φ| 2Φ − 3 16γ|Φ| 4Φ = 0, x ∈ R.
1The terminology soliton was originally used in a context of integrable equations, but we also
The equation (1.5) has nontrivial H1-solutions if and only if (ω, c) satisfies (1.3). For (ω, c) satisfying (1.3), one can rewrite (ω, c) = (ω, 2κ√ω), where the param-eter κ satisfies
−1 < κ ≤ 1 if b > −3/16, −1 < κ < −κ∗ if b ≤ −3/16.
For each parameter κ, the following curve R+3 ω 7→ (ω, 2κ
√
ω) ∈ R2 gives the scaling of the soliton:
φω,2κ√
ω(x) = ω1/4φ1,2κ(
√
ωx) for x ∈ R. When b ≥ 0, there exists a unique κ0 = κ0(b) ∈ (0, 1] such that
E(φ1,2κ0) = P (φ1,2κ0) = 0,
which implies that the soliton uω,2κ0√ω corresponds to the degenerate case. We note that 0 < κ(b) < 1 if b > 0, and κ0(0) = 1. Therefore, algebraic solitons of
(DNLS) correspond to the degenerate case, while degenerate solitons for b > 0 have exponential decay at space infinity. The main purpose of this paper is to establish instability of the degenerate soliton uω,2κ0√ω in a qualitative way.
The degenerate soliton can be also found in a different context, for example, the L2-critical NLS
(NLS) iut= −uxx− |u|4u, (t, x) ∈ R × R,
and the L2-critical generalized KdV equation
(gKdV) ut= −(uxx+ u5)x, (t, x) ∈ R × R.
The equations (NLS) and (gKdV) have the same conserved quantities: E(v) = 1 2kvxk 2 L2 − 1 6kvk 6 L6, (Energy) M(v) = kvk2L2. (Mass)
(NLS) has the standing wave eitQ(x) and (gKdV) has the traveling wave Q(· − t), where Q(x) = 31/4
cosh1/2(2x) is the positive even solution of
−Q00+ Q − Q5= 0, x ∈ R,
and Q is an optimizer of the following Gagliardo–Nirenberg inequality (see [44]): 1 6kf k 6 L6 ≤ 1 2 M(f ) M(Q) 2 kfxk2L2 for f ∈ H1(R). (1.6)
In particular E (Q) = 0 holds, which implies that the solitons eitQ(x) and Q(· − t) correspond to the degenerate case. It is also known that these degenerate solitons are unstable (see [44, 29]).
Instability of degenerate solitons is important to understand the global dynamics of (NLS) and (gKdV). It follows from (1.6) and conservation laws that if the initial data u0 ∈ H1(R) of (NLS) or (gKdV) satisfies M(u0) < M(Q), the corresponding
H1-solution is global and satisfies 1
2 1 −
M(u0)
M(Q) 2!
For (NLS), it is known that finite time blow-up occurs for the initial data satisfying M(u0) > 2π and E (u0) < 0 (see [36]). On the other hand, for (gKdV) existence of
blow-up solutions is a more delicate problem. Martel and Merle [30] proved that finite time blow-up occurs for the initial data satisfying
E(u0) < 0, M(Q) < M(u0) < M(Q) + α0
(1.7)
and some decay condition, where α0 > 0 is a small constant. We note that before
the work [30], the same authors [29] proved instability of the soliton in a qualitative way, which led to an important step for proving the existence of blow-up solutions. For (1.1) for the case b ≥ 0,2 it was proved in [46, 16] that if the initial data u0 ∈ H1(R) satisfies M (u0) < M (φ1,2κ0) =: M
∗, then the corresponding H1
-solution is global and satisfies
kux(t)kL2 ≤ C(ku0kH1) for all t ∈ R,
(1.8)
where the constant in the right-hand side is composed of the conserved quantities E(u0), M (u0), and P (u0). For (DNLS) this mass condition is nothing but the
4π-mass condition. In the recent progress of studies on (DNLS), global well-posedness without the smallness assumption of the mass was established by taking advantage of completely integrable structure (see [40, 20, 2]). These results give a remarkable difference with other L2-critical equations (NLS) and (gKdV), while the uniform boundedness of the flow as (1.8) is not known for M (u0) ≥ 4π. We note that if
we impose further assumptions with M (u0) ≥ 4π, for example, M (u0) = 4π and
P (u0) < 0, or highly oscillating data, then the estimate (1.8) still holds (see [10]).
It was proved in [16] that the mass threshold M∗ gives a certain turning point in variational properties of (1.1). This suggests that global dynamics of (1.1) will change at the mass of M∗. From the variational point of view, M∗ corresponds to the mass threshold M(Q) in (NLS) and (gKdV). Therefore, to investigate the dynamics around the mass of M∗ is important to understand the global dynamics of (1.1). To this end, in this paper we study instability properties of degenerate solitons of (1.1) in a qualitative way.
We first give a precise definition of stability and instability of solitons.
Definition 1.1. We say that the soliton uω,c of (1.1) is stable if for any α > 0
there exists β > 0 such that if u0 ∈ H1(R) satisfies ku0− φω,ckH1 < β, the solution
u(t) of (1.1) exists globally in time and satisfies sup t∈R inf (θ,y)∈R2ku(t) − e iθφ ω,c(· − y)kH1 < α.
Otherwise, we say that the soliton uω,c is unstable.
We now review the known stability results related to our work. When b = 0, Colin and Ohta [6] proved by applying variational approach that if ω > c2/4, the soliton uω,c is stable. For the case c = 2
√
ω some kinds of stability properties were studied in [22, 23], while the stability or instability in the sense of Definition 1.1 remains an open problem. Liu, Simpson and Sulem [27] calculated linearized operators of the generalized derivative nonlinear Schr¨odinger equation
iut+ uxx+ i|u|2σux= 0, (t, x) ∈ R × R, σ > 0,
(gDNLS)
and studied stability of nondegenerate solitons by applying the abstract theory of Grillakis, Shatah and Strauss [12, 13] (see also [14] for partial results in this
direction). Although well-posedness in the energy space for (gDNLS) was assumed in [27], the well-posedness problem was later dealt with in [41, 18, 26].
When b > 0, Ohta [38] proved by applying variational approach in [43, 11, 6] that the soliton uω,cis stable if −2
√ ω < c < 2κ0 √ ω, and unstable if 2κ0 √ ω < c < 2√ω. Ning, Ohta and, Wu [35] proved that the algebraic soliton is unstable for small b > 0, where the assumption of smallness is used for construction of the unstable direction. We note that the momentum of the soliton P (φω,c) is positive in the stable region
{−2√ω < c < 2κ0
√
ω}, negative in the unstable region {2κ0
√
ω < c ≤ 2√ω}, and zero on {c = 2κ0
√
ω} (see Remarks 2 and 3 of [38]). When b < 0 the second author [17] proved by developing variational approaches in [4, 42, 6, 38] that all solitons including algebraic solitons are stable. We note that if b < 0, the momentum of all solitons is positive.
It is known that the stability/instability depends on the spectral properties of the Hessian matrix of the two-variable function
d(ω, c) := Sω,c(φω,c),
where Sω,c is the action defined by
Sω,c(v) := E(v) +
ω
2M (v) + c 2P (v). From a direct computation, we have the identity
det[d00(ω, c)] = √ −2P (φω,c)
4ω − c2{c2+ γ(4ω − c2)} for ω >
c2 4.
If P (φω,c) = 0, then d00(ω, c) has a zero eigenvalue, which corresponds to the
de-generate case.
We note that the abstract theory in [12, 13] is not applicable to degenerate solitons. In [7, 37, 28] instability of degenerate solitons with one-parameter is studied in the abstract framework. The first author [9] extended the work of [37] to degenerate solitons with two-parameter. However, these results are not applicable to degenerate solitons of L2-critical equations (NLS), (gKdV) and (1.1). Recently, Ning [34] proved the instability of the soliton uω,2κ0√
ω of (1.1) for sufficiently small
b > 0. The proof was done by combining localized virial identities and modulation analysis, whose argument was originally developed in [47, 15].
Our approach in the present paper is motivated by the works [29, 8] on instability of degenerate solitons of (gKdV).
We now state our results of this paper. We first organize the spectral properties of the linearized operator around the soliton. The linearized operator is explicitly written as
Lω,cv := Sω,c00 (φω,c)v
(1.9)
= −vxx+ ωv + civx− i|φω,c|2vx− 2i Re(φω,cv)φ0ω,c
− b|φω,c|4v − 4b|φω,c|2Re(φω,cv)φω,c
for v ∈ H1(R). The following claim is used as a basic tool in the proof of our main result.
Proposition 1.2. Let b ∈ R and let (ω, c) satisfy (1.3). Then the space H1(R) is decomposed as the orthogonal direct sum
Here Nω,c is the negative subspace of Lω,c spanned by the eigenvector χω,c
corre-sponding to the simple negative eigenvalue λω,c, Zω,c is the kernel of Lω,c spanned
by iφω,c and φ0ω,c, and Pω,c is the positive subspace of Lω,c such that
(i) if −2√ω < c < 2√ω, then there exists a positive constant k > 0 such that for any p ∈ Pω,c
hLω,cp, pi ≥ kkpk2H1,
(1.10)
(ii) if c = 2√ω, then for any p ∈ Pω,c\ {0}
hLω,cp, pi > 0. (1.11)
We prove Proposition 1.2 by mainly following the argument in [27]. Here we treat the case c = 2√ω, which was not considered in previous works. As in the assertion (ii), the coercivity fails for the case c = 2√ω because the essential spectral of Lω,c consists of the interval [0, ∞) for this case.
We now state our main result, which concerns the instability of degenerate soli-tons of (1.1).
Theorem 1.3. Let b > 0 and c = 2κ0
√
ω and let χω,c be as in Proposition 1.2.
Then there exist α, β ∈ (0, 1) such that if ε0 := u0 − φω,c for the initial data
u0 ∈ H1(R) satisfies
0 < kε0k2H1 ≤ β|(ε0, φω,c)L2|, ε0 ⊥ {χω,c, iφω,c, φ0ω,c, iφ0ω,c},
(1.12)
then there exists t0= t0(u0) ∈ R such that the solution u(t) of (1.1) satisfies
inf
(θ,y)∈R2ku(t0) − e
iθφ
ω,c(· − y)kH1 ≥ α.
In particular, the soliton uω,c is unstable.
Remark 1.4. We can construct ε0 satisfying (1.12) as follows. One can easily show
that the functions χω,c, iφω,c, φ0ω,c, φω,c, iφ0ω,c are linearly independent. Applying
the Gram–Schmidt process, we have a function ε1 ∈ H1(R) satisfying
(ε1, φω,c) 6= 0, ε1 ⊥ {χω,c, iφω,c, φ0ω,c, iφ 0 ω,c}.
Then ε0 := δε1 for small δ > 0 satisfies (1.12).
Remark 1.5. If we replace the assumption (1.12) by 0 < kε0k2H1 ≤ β
(ε0, iφ0ω,c)L2
, ε0 ⊥ {χω,c, iφω,c, φ0ω,c, φω,c}, then the conclusion in Theorem 1.3 still holds.
Remark 1.6. In [34] some explicit function was used as a negative direction of Lω,c
instead of the eigenfunction χω,c. The smallness assumption on b > 0 in [34] comes
from the construction of a negative direction and the explicit formula is also used for the control of modulation parameters. Although one cannot expect the explicit formula of χω,c,3 we construct and control modulation parameters by using the
scaling properties of the equation. Moreover, we obtain a large set of initial data yielding the instability while in [34] the only one unstable direction is found.
3In contexts of (NLS) and (gKdV) one can use the explicit eigenfunction for negative eigenvalue
For the proof of Theorem 1.3 we use modulation theory and the virial identity d dtIm Z xux(t, x)u(t, x) = 4E(u0), (1.13)
but we avoid a direct use of this identity. We consider the decomposition u(t, x) = e iθ(t) λ(t)1/2 (φω,c+ ε) t,x − x(t) λ(t) , (1.14)
where λ(t) > 0, θ(t) ∈ R, x(t) ∈ R, and the function ε(t, x) satisfies suitable orthogonal conditions (see Proposition 3.2). If we put the formula (1.14) into (1.13), the left-hand side of (1.13) yields the quantity
d dtIm Z ε(t, x)Λφω,c(x) Λf := f2 + xfx , (1.15)
which plays an essential role in our proof. This quantity has already been effectively used on the studies of the blow-up dynamics of (NLS) (see, e.g., [31]), but it seems to be new in the contexts of (1.1), (DNLS) and (gDNLS). The quantity (1.15) is well-defined in the H1-setting, so we do not need any cut-off arguments, which becomes a much simpler argument than previous works [47, 15, 34]. Moreover, our proof gives a close relation to instability theory on (gKdV) (see Appendix A).
The arguments except one step (Lemma 3.7) in our proof work for the case b = 0 and c = 2√ω, i.e., algebraic solitons of (DNLS), in exactly the same way.4 Although we could not complete the proof of Theorem 1.3 for the case b = 0, the unstable directions are detected in the same way as the case b > 0 (see Lemma 4.3). Therefore, we believe that the conclusion of Theorem 1.3 is still true for algebraic solitons of (DNLS).
In the assumption of Theorem 1.3, if we consider the initial data u0 = φω,c+ ε0
with (ε0, φω,c)L2 > 0, then
(1.16) E(u0) < 0, M (φω,c) < M (u0) < M (φω,c) + β0,
where β0 is a small constant. We note that the condition (1.16) corresponds to the
blow-up set of (NLS) and (gKdV), and so Theorem 1.3 gives an important clue to construct a singular solution of (1.1).
The rest of this paper is organized as follows. In Section 2 we study the spectra of the linearized operator Lω,c and prove Proposition 1.2. In Section 3 we construct
the modulation parameters satisfying suitable orthogonal conditions and control these parameters. In Section 4 we organize the virial identities. In Section 5 we complete the proof of Theorem 1.3 by using the estimates obtained in previous sections.
2. Structure of the linearized operator
In this section, we study the structure of the linearized operator Lω,c.
Through-out this section, we assume that (ω, c) satisfies (1.3). For simplicity we often drop the subscript (ω, c) as
S = Sω,c, φ = φω,c, Φ = Φω,c.
4For the proof of Lemma 3.7, we use the coercivity property of L
ω,cwhich does not hold in the
We define the function ηω,c as η(x) = ηω,c(x) = c 2x − 1 4 Z x −∞ Φω,c(y)2dy, (2.1)
and define the operator ˜Lω,c as
˜
L = ˜Lω,c = e−iηω,c(x)Lω,ceiηω,c(x).
For w ∈ H1(R) we set f = Re w and g = Im w. After a direct calculation, ˜Lw is explicitly represented as ˜ Lw = −wxx+ ω − c 2 4 w +c 2Φ 2w + cΦ2Re w − 3 16γΦ 4w −3 4γΦ 4Re w (2.2) + 1 4Φ 4Re w − i 2Φ 2w x+ i 2ΦΦ 0w − 2iΦΦ0Re w = L11f + L12g + 1 4Φ 4f + i(L 21f + L22g), where L11:= −∂x2+ UΦ, UΦ := ω − c 2 4 +3 2cΦ 2− 15 16γΦ 4, L12:= 1 2Φ 2∂ x− 1 2ΦΦ 0, L21:= − 1 2Φ 2∂ x− 3 2ΦΦ 0, L22:= −∂x2+ VΦ, VΦ := ω − c 2 4 +c 2Φ 2− 3 16γΦ 4.
Since eiη(x) is a unitary operator, the spectral property of ˜L is the same as that of
L. In what follows, we investigate the spectra of the operator ˜L.
We first note that ˜L can be considered as compact perturbation of the operator −∂2
x+ (ω − c2/4). Therefore, by Weyl’s theorem we deduce that
σess( ˜L) = σess −∂2 x+ ω −c 2 4 =hω − c 2 4, ∞
and the spectrum of ˜L in (−∞, ω − c2/4) consists of isolated eigenvalues.
2.1. Kernel. In this subsection we prove the nondegeneracy of the kernel of ˜L. Our proof depends on the argument in [25].
Lemma 2.1. The following statement is true. (i) ker L11= span{Φ0ω,c},
(ii) ker L22= span{Φω,c}.
Proof. Since Φ is a solution of (1.5), we have L22Φ = 0. By differentiating the
equation (1.5), we also have L11Φ0 = 0. Hence we have
ker L11⊃ span{Φ0}, ker L22⊃ span{Φ}.
It now suffices to show ker L22⊂ span{Φ} because one can show ker L11⊂ span{Φ0}
by the same argument. Let g ∈ ker L22. We consider the Wronskian of Φ and g:
From Φ, g ∈ H2(R), we have W (x) → 0 as |x| → 0. Since L22Φ = L22g = 0, we
obtain
W0(x) = Φ00g − Φg00= VΦΦg − ΦVΦg = 0.
Thus, we deduce W ≡ 0, which implies that Φ and g are linearly dependent. This
completes the proof.
Lemma 2.2. The kernel ˜Lω,c is determined by
ker ˜Lω,c = span iΦω,c, Φ0ω,c− i 4Φ 3 ω,c , which is equivalent to ker Lω,c = span{iφω,c, φ0ω,c}.
Proof. First we show ker ˜L ⊃ spaniΦ, Φ0−4iΦ3 . Since φ is a solution of (1.4), and the equation has symmetries under the phase and spatial translation, we have S0(eiθφ(· − y)) = 0 for all (θ, y) ∈ R × R. Differentiating this with respect to θ or y at (θ, y) = 0, we have
(2.3) Liφ = 0, Lφ0= 0,
respectively. Since e−iη(x)L = ˜Le−iη(x) and φ = eiη(x)Φ, (2.3) is equivalent to ˜ LiΦ = 0, L˜ Φ0+ ic 2Φ − i 4Φ 3= 0.
This implies ker ˜L ⊃ spaniΦ, Φ0−4iΦ3 .
Next we show the inverse inclusion. Let w ∈ ker ˜L, f = Re w, and g = Im w. The expression (2.2) of ˜L implies that (f, g) satisfies the following system of ordinary differential equations: L11f + L12g + 1 4Φ 4f = 0, L21f + L22g = 0. (2.4)
Now we apply the following transformation to g: g = h −1 2Φ Z x −∞ Φf dy. (2.5) Then we have L12g + 1 4Φ 4f = 1 2Φ 2g x− 1 2ΦΦ 0 g +1 4Φ 4f (2.6) = 1 2Φ 2h x− 1 2ΦΦ 0h.
Moreover, noting that ∂x2 1 2Φ Z x −∞ Φf dy = 1 2Φ 00Z x −∞ Φf dy + 3 2ΦΦ 0f + 1 2Φ 2f x = 1 2Φ 00Z x −∞ Φf dy − L21f,
it follows from L22Φ = 0 that
L21f + L22g = L21f + L22h + ∂x2 1 2Φ Z x −∞ Φf dy −1 2VΦΦ Z x −∞ Φf dy (2.7) = L22h − 1 2(−Φ 00 + VΦΦ) Z x −∞ Φf dy = L22h.
Using (2.6) and (2.7) we write the equation (2.4) as L11f + 1 2Φ Φhx− Φ 0h = 0, L22h = 0. (2.8)
From the second equation in (2.8) and Lemma 2.1 (ii), we have h = αΦ for some α ∈ R. Substituting this into the first equation in (2.8), we get L11f = 0. Therefore,
Lemma 2.1 (i) implies that f = βΦ0 for some β ∈ R. Substituting h = αΦ and f = βΦ0 into (2.5), we have g = αΦ −β 2Φ Z x −∞ ΦΦ0dy = αΦ −β 4Φ Z x −∞ (Φ2)0dy = αΦ −β 4Φ 3.
Therefore, we obtain that
w = f + ig = βΦ0+ i αΦ − β 4Φ 3 = αiΦ + β Φ0− i 4Φ 3 ∈ span iΦ, Φ0− i 4Φ 3 .
This completes the proof.
2.2. Construction of a negative direction. In this subsection we prove that ˜
Lω,c has exactly one negative eigenvalue. Our proof depends on the argument in
[27] (see also [14]). The following expression of the quadratic form is useful to construct a negative direction.
Lemma 2.3. Let w ∈ H1(R), f = Re w, and g = Im w. Then we have h ˜Lω,cw, wi = hL11f, f i + 1 4kΦ 2 ω,cf + 2Φω,c∂x(Φ−1ω,cg)k2L2. (2.9)
Proof. First, by the expression (2.2), we have h ˜Lw, wi = hL11f, gi + hL12g, f i +
1 4hΦ
4f, f i + hL
21f, gi + hL22g, gi.
We set ˜g = Φ−1g. It follows from L22Φ = 0 that
hL22g, gi = h˜g(−∂x2+ VΦ)Φ, Φ˜gi − h2Φ0g˜x+ Φ˜gxx, Φ˜gi
= −h∂x(Φ2˜gx), ˜gi = kΦ˜gxk2L2.
Next, we calculate the interaction terms as hL12g, f i =D1 2Φ 2∂ x(Φ˜g) − 1 2Φ 2Φ0g, f˜ E= 1 2hΦ 3, f ˜g xi and hL21f, gi = − 1 2Φ 2f x+ 3 2ΦΦ 0f, Φ˜g = −1 2hΦ 3, ˜gf xi − 1 2h∂x(Φ 3), f ˜gi = 1 2hΦ 3, f ˜g xi.
Therefore we deduce that
h ˜Lw, wi = hL11f, gi + 1 4hΦ 4f, f i + hΦ3, f ˜g xi + kΦ˜gxk2L2 = hL11f, f i + 1 4kΦ 2f + 2Φ˜g xk2L2.
This completes the proof. Lemma 2.4. The operator L11 has exactly one negative eigenvalue.
Proof. We note that L11is a compact perturbation of the operator −∂x2+(ω −c2/4).
Therefore, by Weyl’s theorem we deduce that σess(L11) = σess −∂x2+ ω − c 2 4 = h ω −c 2 4, ∞ ,
and the spectrum of L11in (−∞, ω − c2/4) consists of isolated eigenvalues. We note
that L11Φ0= 0 and that Φ0 has exactly one zero point. By Sturm–Liouville theory
we deduce that zero is the second eigenvalue of L11, and that L11has one negative
eigenvalue. Moreover, one can prove that the negative eigenvalue is simple (see, e.g., [1, Theorem B.59]). This completes the proof. We denote the negative eigenvalue of L11in Lemma 2.4 by λ11and its normalized
eigenvector by χ11, that is,
(2.10) L11χ11= λ11χ11, kχ11kL2 = 1.
Lemma 2.5. The operator ˜Lω,c has exactly one negative eigenvalue.
Proof. Let χ12:= − 1 2Φ Z x −∞ Φχ11dy. Then we have Φ∂x(Φ−1χ12) = − 1 2Φ 2χ 11.
Therefore, it follows from (2.9) and (2.10) that χ∗ := χ11+ iχ12 satisfies
h ˜Lχ∗, χ∗i = hL11χ11, χ11i = λ11< 0.
This means that the operator ˜L has at least one negative eigenvalue.
Now we show that ˜L has exactly one negative eigenvalue. Assume that ˜L has two negative eigenvalues (including repeats) λ1 ≤ λ2< 0 with eigenvectors χ1 and
χ2 such that
˜
Lχ1 = λ1χ1, Lχ˜ 2 = λ2χ2, kχ1kL2 = kχ2kL2 = 1, (χ1, χ2)L2 = 0.
We note that by the formula (2.9) and Lemma 2.4, h ˜Lp, pi ≥ 0 for each p ∈ H1(R) satisfying (Re p, χ
11)L2 = 0. Thus, it follows from h ˜Lχ2, χ2i = λ2 < 0 that
(Re χ2, χ11)L2 6= 0. If we set
α = −(Re χ1, χ11)L2 (Re χ2, χ11)L2
, p0= χ1+ αχ2,
then we have (Re p0, χ11)L2 = 0. Hence we deduce that h ˜Lp0, p0i ≥ 0. On the other
hand, by a direct calculation we obtain
h ˜Lp0, p0i = λ1+ α2λ2 < 0,
which yields a contradiction. This completes the proof. Remark 2.6. When b ≥ 0, by variational characterization of the solitons (see [6, 10, 16]) one can prove that Lω,c has exactly one negative eigenvalue (see the argument
of [25]). Our approach based on the formula (2.9) is more elementary and applicable to the case b < 0 in a unified way.
2.3. Spectral decomposition. We now complete the proof of Proposition 1.2. Proof of Proposition 1.2. By Lemma 2.2 and Lemma 2.5, we have the following decomposition H1(R) = span{ ˜χ} ⊕ span iΦ, Φ0− i 4Φ 3 ⊕ ˜P, (2.11)
where ˜χ is the eigenvector of ˜L corresponding to its negative eigenvalue λ and ˜P is the nonnegative subspace of ˜L. Since ˜L = e−iη(x)Leiη(x), (2.11) is equivalent that
H1(R) = N ⊕ Z ⊕ P, (2.12)
where N is spanned by the negative eigenvector χ := eiη(x)χ of L, Z := span{iφ, φ˜ 0}
is its kernel, and P := eiη(x)P is its nonnegative subspace. The rest of the proof is˜ to show the positivity of L on P.
(i) We consider the case −2√ω < c < 2√ω. Since σess(L) =
h
ω − c2/4, ∞, the spectra of L except for its negative eigenvalue and zero eigenvalue are positive and bounded away from zero. Therefore, there exists a positive constant δ0 > 0 such
that
hLp, pi ≥ δ0kpk2L2 for all p ∈ P.
(2.13)
From the explicit formula (1.9), there exists a positive constant C0 such that
hLv, vi ≥ 1 2kvxk
2
L2− C0kvk2L2
for all v ∈ H1(R). Combined with (2.13), we have kpk2H1 ≤ 2hLp, pi + (1 + 2C0)kpk2L2 ≤ 2 +1 + 2C0 δ0 hLp, pi for all p ∈ P, which shows the desired inequality (1.10).
(ii) We now consider the case c = 2√ω. Assume by contradiction that there exists p0 ∈ P such that kp0kL2 = 1 and hLp0, p0i = 0. Then we obtain the following
relation:
hLp0, p0i = min{hLp, pi : kpkL2 = 1, (χ, p)L2 = (iφ, p)L2 = (φ0, p)L2 = 0}.
This minimization problem implies that there exist Lagrange multipliers α1, α2,
α3, and α4 such that
Lp0= α1χ + α2iφ + α3φ0+ α4p0.
By the orthogonal conditions and hLp0, p0i = 0, we have α1 = α2 = α3 = α4 = 0.
Therefore, p0 ∈ ker L ∩ P = {0}, which is a contradiction. Hence (1.11) holds.
3. Modulation theory
In this section we organize modulation theory for three fundamental symmetries which are phase, translation, and scaling.
We prepare some notations. For α > 0 we define a tubular neighborhood around the soliton φω,c by
Uα = {u ∈ H1(R) : inf (θ,z)∈R2ke
iθu(· + z) − φ
For u ∈ H1(R), λ > 0, and θ, y ∈ R, we denote the function ε by ε(λ, θ, x; u) = ε(λ, θ, x) = λ1/2e−iθu(λ · +x) − φω,c.
For λ > 0 and f : R → C, we define the rescaling fλ(y) = λ1/2f (λy). Let Λ be the generator of this transformation as
Λf := ∂λfλ|λ=1=
f
2 + yfy. We note that Λ is skew-symmetric, i.e.,
(Λf, g)L2 = −(f, Λg)L2.
3.1. Construction of modulation parameters. We construct the modulation parameters λ, θ, and x satisfying suitable orthogonal conditions. We first prepare the following lemma.
Lemma 3.1. Assume that (ω, c) satisfy (1.3). Then we have (i) (Λφω,c, iφω,c)L2 = (Λφω,c, φ0ω,c)L2 = 0.
If we further assume b ≥ 0 and c = 2κ0(b)
√
ω, then we have (ii) (iφ0ω,c, Λφω,c)L2 = (iφ0ω,c, φω,c)L2 = 0,
(iii) (Λφω,c, χω,c)L2 6= 0.
Proof. (i) It follows from the explicit formula of η (see (2.1)) that η0 = c
2− 1 4Φ
2,
φ0 = eiη iη0Φ + Φ0 = eiη
ic 2Φ − i 4Φ 3+ Φ0 . Since Φ is a real-valued and even function, one computes easily that
(Λφ, iφ)L2 = φ 2 + yφ 0, iφ L2 = (yφ 0, iφ) L2 = Re Z y ic 2Φ − i 4Φ 3+ Φ0 · (−iΦ) = Re Z y c 2Φ 2−1 4Φ 4 = 0, (Λφ, φ0)L2 = φ 2 + yφ 0 , φ0 L2 = (yφ0, φ0)L2 = Re Z y (Φ0)2+ c 2Φ − 1 4Φ 3 2 = 0. (ii) Since P (φ) = 0 by c = 2κ0 √ ω, we have
(iφ0, Λφ)L2 = (iφ0,φ2 + yφ0)L2 = Re
Z
iy|φ0|2 = 0.
(iii) By twice differentiating the following relation
S(φλ) = λ2E(φ) + ωM (φ) + λcP (φ)
at λ = 1, we have hLΛφ, Λφi = 2E(φ) = 0. Suppose that (Λφ, χ)L2 = 0. From
Proposition 1.2 and (i) proved just above, we obtain that hLΛφ, Λφi > 0. This is a
contradiction and completes the proof.
Proposition 3.2. Let b ≥ 0 and c = 2κ0
√
ω. Then there exist constants α0 >
0, λ0 > 0 and C1-mappings (Λ, Θ, X) : Uα0 → (1 − λ0, 1 + λ0) × R
2 such that
(ε(Λ(u), Θ(u), X(u)), χω,c)L2 = (ε(Λ(u), Θ(u), X(u)), iφω,c)L2
= (ε(Λ(u), Θ(u), X(u)), φ0ω,c)L2 = 0
(3.1)
for all u ∈ Uα0. Moreover, there exists a constant C > 0 such that for any α ∈
(0, α0) and u ∈ Uα
kε(Λ(u), Θ(u), X(u))kH1 ≤ Cα, |Λ(u) − 1| ≤ Cα.
(3.2)
Proof. Let F : (0, ∞) × R2× H1(R) → R be the function defined by
F (λ, θ, x; u) = (ε(λ, θ, x; u), χ)L2 (ε(λ, θ, x; u), iφ)L2 (ε(λ, θ, x; u), φ0)L2 .
We define the open neighborhoods Vα of φ and Ωδ⊂ (0, ∞) × R2 of (1, 0, 0) by
Vα = {u ∈ H1(R) : ku − φkH1 < α},
Ωδ = {(λ, θ, x) ∈ (0, ∞) × R2: |λ − 1| + |θ| + |x| < δ}.
By the orthogonality between ker L and χ, and Lemma 3.1, we have ∂F
∂(λ, θ, x)(1, 0, 0; φ) =
(Λφ, χ)L2 −(iφ, χ)L2 (φ0, χ)L2
(Λφ, iφ)L2 −(iφ, iφ)L2 (φ0, iφ)L2
(Λφ, φ0)L2 −(iφ, φ0)L2 (φ0, φ0)L2 = (Λφ, χ)L2 0 0 0 −kφk2 L2 0 0 0 kφ0k2 L2 .
Since (Λφ, χ)L2 6= 0 by Lemma 3.1 (3), we deduce that
det ∂F
∂(λ, θ, x)(1, 0, 0; φ) 6= 0. (3.3)
Combined with F (1, 0, 0; φ) = 0, the implicit function theorem implies that there exist constants ¯α > 0 and ¯δ > 0 and C1-mappings (Λ, Θ, X) : V
¯
α→ Ωδ¯such that
F (Λ(u), Θ(u), X(u); u) = 0 for all u ∈ Vα¯
(3.4) and
|Λ(u) − 1| + |Θ(u)| + |X(u)| . ku − φkH1 for all u ∈ Vα¯.
(3.5)
By the expression of ε(Λ(u), Θ(u), X(u)) and (3.5), one can compute easily that kε(Λ(u), Θ(u), X(u))kH1 . ku − φkH1 for u ∈ Vα¯.
In particular, for α ∈ (0, ¯α) we have
kε(Λ(u), Θ(u), X(u))kH1 . α, |Λ(u) − 1| . α for u ∈ Vα.
(3.6)
By possibly choosing α smaller, we can extend Λ, Θ, and X to the functions defined on the tubular neighborhood Uα (see, e.g., [24] for more details). This completes
3.2. Control of the modulation parameters. Now we derive the equation for ε and estimate on the modulation parameters.
Let u0 ∈ Uα0 and u(t) be the solution of (1.1) with u(0) = u0. We denote the
exit times from the tubular neighborhood Uα by
Tα±= inf{t > 0 : u(±t) /∈ Uα}.
We set Iα= (−Tα−, Tα+). Since u(t) ∈ Uα0 for t ∈ Iα0, we can define
λ(t) = Λ(u(t)), θ(t) = Θ(u(t)), x(t) = X(u(t)), (3.7)
where the functions (Λ, Θ, X) are given in Proposition 3.2. We see that λ(t), θ(t), and x(t) are C1-functions on Iα0. For t ∈ Iα0 we denote
v(t) = v(t, y) = λ(t)1/2e−iθ(t)u(t, λ(t)y + x(t)) (3.8)
and define the function ε(t) by
ε(t) = ε(λ(t), θ(t), x(t); u(t)) = v(t) − φω,c.
(3.9)
We rescale the time as follows. We set ˜ s(t) = Z t 0 dτ λ(τ )2, I˜α0 = ˜s(Iα0).
Obviously t 7→ ˜s(t) is strictly increasing, so the inverse function ˜t := ˜s−1 exists. For a function Iα0 3 t 7→ f (t), we define ˜Iα0 3 s 7→ ˜f (s) by
˜ f (s) = f (˜t(s)). We note that ˜ fs(s) = ft(t)λ(t)2 for s = ˜s(t). (3.10)
For simplicity of notations, in what follows we omit “tilde” over the functions of the variable s although it is the same symbol as the function of the variable t. Lemma 3.3. For s ∈ Iα0, ε(s) satisfies
iεs= Lε + (θs− ω)φω,c+ xs λ − c iφ0ω,c+λs λiΛφω,c (3.11) + (θs− ω)ε + xs λ − c iεy+ λs λiΛε + R(ε),
where R(ε) is the sum of second and higher order terms of ε explicitly written as R(ε) = −i|ε|2φ0ω,c− 2i Re(εφω,c)εy− 4b{Re(εφω,c)}2φω,c− 2b|φω,c|2|ε|2φω,c
− 4b|φω,c|2Re(εφ
ω,c)ε − i|ε|2εy− 4b|ε|2Re(εφω,c)φω,c− 4b{Re(εφω,c)}2ε
− 2b|φω,c|2|ε|2ε − b|ε|4φω,c− 4b|ε|2Re(εφω,c)ε − b|ε|4ε,
and there exists C > 0 such that (3.12)
Z
|R(ε)| ≤ C(kεk2L2+ kεkL2kεykL2) for ε ∈ H1(R) with kεkH1 ≤ 1.
Proof. By direct calculations we see that v(t) satisfies the equation iλ2vt= −vyy− i|v|2vy− b|v|4v + λtλiΛv + θtλ2v + xtλivy.
By rescaling the time and (3.10), we have ivs= −vyy− i|v|2vy− b|v|4v +
λs
λiΛv + θsv + xs
By substituting v(s) = φ + ε(s), we obtain that iεs= ivs= −vyy− i|v|2vy− b|v|4v + λs λiΛv + θsv + xs λivy (3.13) = −(φ + ε)yy− i|φ + ε|2(φ + ε)y− b|φ + ε|4(φ + ε) +λs λiΛ(φ + ε) + θs(φ + ε) + xs λi(φ + ε)y. We now set
R1(ε) = −i|φ + ε|2(φ + ε)y+ i|φ|2φ0+ i|φ|2εy+ 2i Re(εφ)φ0
= −i|ε|2φ0− 2i Re(εφ)εy − i|ε|2εy,
R2(ε) = −b|φ + ε|4(φ + ε) + b|φ|4φ + b|φ|4ε + 4b|φ|2Re(εφ)φ
= −b
4{Re(εφ)}2φ + |ε|4φ + 4|ε|2Re(εφ)φ + 2|φ|2|ε|2φ
+ 4{Re(εφ)}2ε + |ε|4ε + 4|φ|2Re(εφ)ε + 4|ε|2Re(εφ)ε + 2|φ|2|ε|2ε
, and R(ε) = R1(ε) + R2(ε). By the Sobolev embedding we have
Z
(|R1(ε)| + |R2(ε)|) . kεk2L2+ kεkL2kεykL2 for ε ∈ H1(R) with kεkH1 ≤ 1.
From (3.13), we obtain that
iεs= −(φ + ε)yy+ R1(ε) − i|φ|2φ0− i|φ|2εy− 2i Re(εφ)φ0
+ R2(ε) − b|φ|4φ − 3b|φ|4ε − 2b|φ|2φ2ε + λs λiΛ(φ + ε) + θs(φ + ε) + xs λi(φ + ε)y = −εyy− i|φ|2εy − 2i Re(εφ)φ0− 3b|φ|4ε − 2b|φ|2φ2ε − φ00− i|φ|2φ0− b|φ|4φ +λs λiΛ(φ + ε) + θs(φ + ε) + xs λi(φ + ε)y+ R(ε). By using the relations
− εyy− i|φ|2εy− 2i Re(εφ)φ0− 3b|φ|4ε − 2b|φ|2φ2ε = Lε − ωε − ciεy,
− φ00− i|φ|2φ0− b|φ|4φ = −ωφ − ciφ0,
we obtain (3.11).
We note that from Proposition 3.2,
(ε(s), χω,c)L2 = (ε(s), iφω,c)L2 = (ε(s), φ0ω,c)L2 = 0,
(3.14)
kε(s)kH1 ≤ Cα, |λ(s) − 1| ≤ Cα
(3.15)
hold for α ∈ (0, α0) and s ∈ Iα, where C is independent of α and s.
Lemma 3.4. Let b ≥ 0 and c = 2κ0
√
ω. For s ∈ Iα0, the following equalities hold.
λs λ(Λφω,c, χω,c)L2 = −(ε, Lω,ciχω,c)L2 − (θs− ω)(ε, iχω,c)L2 + xs λ − c (ε, χ0ω,c)L2+ λs λ(ε, Λχω,c)L2 − (R(ε), iχω,c)L2, (θs− ω)kφω,ck2L2 = −(ε, Lω,cφω,c)L2 − (θs− ω)(ε, φω,c)L2
−xs λ − c (ε, iφ0ω,c)L2− λs λ(ε, Λiφω,c)L2− (R(ε), φω,c)L2, xs λ − c kφ0ω,ck2L2 = −(ε, Lω,ciφω,c0 )L2 − (θs− ω)(ε, iφ0ω,c)L2 + xs λ − c (ε, φ00ω,c)L2 + λs λ(ε, Λφ 0 ω,c)L2 − (R(ε), iφ0ω,c)L2.
Moreover, there exist C > 0 and α1 ∈ (0, α0) such that for s ∈ Iα1, the following
estimate holds. λs λ + |θs− ω| + xs λ − c ≤ Ckε(s)kL2. (3.16)
Proof. By differentiating the orthogonal relation (ε(s), χ)L2 = 0 with respect to s,
we have the first relation in the statement as follows: 0 = (εs, χ)L2 = −(iLε, χ)L2 − (θs− ω)(iφ, χ)L2 + xs λ − c (φ0, χ)L2+ λs λ(Λφ, χ)L2 − (θs− ω)(iε, χ)L2+ xs λ − c (εy, χ)L2+ λs λ(Λε, χ)L2 − (iR(ε), χ)L2 = (ε, Liχ)L2 + λs λ(Λφ, χ)L2 + (θs− ω)(ε, iχ)L2− xs λ − c (ε, χ0)L2 − λs λ(ε, Λχ)L2 + (R(ε), iχ)L2, where we used (iφ, χ)L2 = (φ0, χ)L2 = 0 in the last equality.
From Lemma 3.1 we recall that the following equalities hold. (Λφ, iφ)L2 = (Λφ, φ0)L2 = (iφ0, φ)L2 = 0.
By differentiating the relation (ε(s), iφ)L2 = 0 with respect to s, we obtain the
second relation as 0 = (εs, iφ)L2
= −(iLε, iφ)L2 − (θs− ω)(iφ, iφ)L2+
xs λ − c (φ0, iφ)L2 + λs λ(Λφ, iφ)L2 − (θs− ω)(iε, iφ)L2 + xs λ − c (εy, iφ)L2 + λs
λ(Λε, iφ)L2 − (iR(ε), iφ)L2 = −(ε, Lφ)L2 − (θs− ω)kφk2L2 − (θs− ω)(ε, φ)L2 − xs λ − c (ε, iφ0)L2− λs λ(ε, iΛφ)L2− (R(ε), φ)L2. Similarly, by differentiating the relation (ε(s), φ0)L2 = 0 with respect to s, we obtain
the third relation as 0 = (εs, φ0)L2 = − (iLε, φ0)L2− (θs− ω)(iφ, φ0)L2+ xs λ − c (φ0, φ0)L2 + λs λ(Λφ, φ 0) L2 − (θs− ω)(iε, φ0)L2+ xs λ − c (εy, φ0)L2 + λs λ(Λε, φ 0) L2 − (iR(ε), φ0)L2 = (ε, Liφ0)L2+ xs λ − c kφ0k2L2
+ (θs− ω)(ε, iφ0)L2− xs λ − c (ε, φ00)L2 − λs λ(ε, Λφ 0 )L2 + (R(ε), iφ0)L2.
From three relations above and (3.12), we obtain λs λ + |θs− ω| + xs λ − c .kεkL2+ λs λ + |θs− ω| + xs λ − c kεkL2.
By (3.15) and taking α small enough, we obtain the estimate (3.16). 3.3. Error estimates. In this subsection, we derive the uniform estimate of ε(s) for s ∈ Iα0. Assume that ε0∈ H
1(R) satisfies
(ε0, χω,c)L2 = (ε0, iφω,c)L2 = (ε0, φ0ω,c) = 0.
(3.17)
We set u0 = φω,c+ ε0. From (3.4) and (3.17), we have
λ(0) = Λ(u0) = 1, θ(0) = Θ(u0) = 0, x(0) = X(u0) = 0,
which implies that
ε(0) = ε(λ(0), θ(0), x(0)) = ε(1, 0, 0) = u0− φω,c = ε0. We define Ee(ε) = E(φω,c+ ε) − E(φ), Me(ε) = M (φω,c+ ε) − M (φω,c) = 2(φω,c, ε)L2+ M (ε), Pe(ε) = P (φω,c+ ε) − P (φω,c) = 2(iφ0ω,c, ε) + P (ε), Se(ε) = Sω,c(φ + ε) − Sω,c(φ) = Ee(ε) + ω 2Me(ε) + c 2Pe(ε). Lemma 3.5. For ε ∈ H1(R), we have
Ee(ε) = −ω(φω,c, ε)L2 − c(iφω,c0 , ε)L2+ O(kεk2H1), Me(ε) = 2(φω,c, ε)L2+ O(kεk2H1), Pe(ε) = 2(iφ0ω,c, ε)L2 + O(kεk2H1), Se(ε) = 1 2hLω,cε, εi + O(kεk 3 H1) = O(kεk2H1).
Proof. Since S0(φ) = 0, this is equivalent to
E0(φ) = −ωφ − ciφ0. By the Taylor expansion we have
Ee(ε) = E(φ + ε) − E(φ) = hE0(φ), εi + O(kεk2H1)
= −ω(φ, ε)L2− c(iφ0, ε)L2 + O(kεk2H1), Se(ε) = S(φ + ε) − S(φ) = 1 2hLε, εi + O(kεk 3 H1).
The estimates for Me and Pe are trivial from the definition.
Lemma 3.6. Let b ≥ 0 and c = 2κ0
√
ω. For s ∈ Iα0, we have
Proof. A direct computation shows that
M (φ + ε(s)) = M (v(s)) = M (u(s)) = M (u0) = M (φ + ε0).
By expanding both sides we deduce that
2(φ, ε(s))L2 + M (ε(s)) = 2(φ, ε0)L2+ M (ε0),
which is the desired equality.
Since E(φ) = P (φ) = 0 from the assumption, we have
Ee(ε(s)) = E(φω,c+ ε(s)) = E(v(s)), Pe(ε(s)) = P (φω,c+ ε(s)) = P (v(s)).
Therefore, we deduce that
Pe(ε(s)) = P (v(s)) = λ(s)P (u(t(s))) = λ(s)P (u0) = λ(s)Pe(ε0),
Ee(ε(s)) = E(v(s)) = λ(s)2E(u(t(s))) = λ(s)2E(u0) = λ(s)2Ee(ε0).
This completes the proof.
Lemma 3.7. Let b > 0 and c = 2κ0
√
ω. Then there exist C > 0 and α2 ∈ (0, α0)
such that for any α ∈ (0, α2) and s ∈ Iα, we have
kε(s)k2H1 ≤ C α|2ω(φω,c, ε0)L2 + c(iφ0ω,c, ε0)L2|
(3.18)
+ α2|ω(φω,c, ε0)L2 + c(iφ0ω,c, ε0)L2| + kε0k2H1.
Proof. Since ω > c2/4 from the assumption, we note that the coercivity property (1.10) holds. It follows from Lemma 3.5 and (3.15) that by taking α small enough,
Se(ε(s)) =
1
2hLε(s), ε(s)i + O(kε(s)k
3
H1) & kε(s)k2H1.
On the other hand, we deduce from Lemmas 3.5 and 3.6 that Se(ε(s)) = λ(s)2Ee(ε0) + ω 2Me(ε0) + λ(s) c 2Pe(ε0) = Se(ε0) + (λ(s)2− 1)Ee(ε0) + (λ(s) − 1) c 2Pe(ε0) = (λ(s) − 1) 2Ee(ε0) + c 2Pe(ε0) + (λ(s) − 1)2Ee(ε0) + O(kε0k2H1) = (λ(s) − 1) −2ω(φ, ε0)L2 − c(iφ0, ε0)L2 − (λ(s) − 1)2 ω(φ, ε0)L2 + c(iφ0, ε0)L2 + O(kε0k2H1).
Therefore, combined with (3.15), we obtain (3.18).
4. Virial identities
In this section we organize virial identities of (1.1). Let u be the H1-solution of (1.1) with u(0) = u0 ∈ H1(R), which is defined on a maximal interval (−Tmin, Tmax).
Proposition 4.1 (Virial identity). For u0 ∈ H1(R) such that R x2|u0|2 < ∞, we
have the following relations: d dt Z x2|u|2 = 4 Im Z xuxu + Z x|u|4, (4.1) d dtIm Z xuxu = 4E(u0) (4.2)
Proof. See [45, Lemma 2.2] and [3, Proposition 6.5.1]. The first relation (4.1) is different from the one of (NLS) due to the appearance of the second term in the right-hand side. On the other hand, the second relation (4.2) is the same as (NLS). We take advantage of the latter relation for the proof of instability.
We now assume that u(0) = u0 ∈ Uα0. We recall that v(t) and ε(t) are defined
in (3.8) and (3.9), respectively. We rescale the time variable t to s as in Section 3. Following [31], we rewrite the virial relation in terms of ε(s). We denote
J [v] = Im Z
yvyv dy = − Re
Z
iyvyv dy.
Then J [ε] is represented as follows. Lemma 4.2. Let b ≥ 0 and c = 2κ0
√ ω. Assume thatR x2|u 0|2< ∞. For s ∈ Iα0, we have J [ε(s)] = 2(ε(s), iΛφω,c)L2+ J [u(s)] + x(s)P (u0). (4.3)
Proof. From the phase and scaling invariance of J , we have J [v(s)] = J [u(s, · + x(s)] = J [u(s)] + x(s)P (u0).
(4.4)
On the other hand, J [v(s)] is rewritten as
J [v(s)] = J [ε(s) + φ] = J [ε(s)] − 2(ε, iΛφ)L2 + J [φ].
(4.5)
By Lemma 3.1, J [φ] is rewritten as J [φ] = (iφ, yφ0)L2 = (iφ,1
2φ + yφ 0)
L2 = (iφ, Λφ)L2 = 0.
(4.6)
By combining (4.4), (4.5), and (4.6), we obtain (4.3). The first term in the right-hand side of (4.3)
(ε(s), iΛφω,c)L2 = Im
Z
ε(s)Λφω,c
(4.7)
plays an essential role in our proof of instability. We note that (4.7) is well-defined without the assumption R x2|u
0|2 < ∞. From the equation (3.11), we have
d ds(ε(s), iΛφ)L2 = −(iεs(s), Λφ)L2 = − Lε + (θs− ω)φ + i xs λ − c φ0+ iλs λΛφ + (θs− ω)ε + i xs λ − c εy+ i λs λΛε + R(ε), Λφ L2
for s ∈ Iα0. We note that (φ, Λφ)L2 = (iΛφ, Λφ)L2 = 0 and (iφ
0, Λφ)
L2 = 0 by
Lemma 3.1 (2). Therefore, by (3.12) and (3.16), we deduce that d
ds(ε(s), iΛφ)L2 = −(ε(s), LΛφ)L2+ O(kε(s)k
2 H1)
(4.8)
for s ∈ Iα1, where α1> 0 appeared in Lemma 3.4. Therefore, by using the relation
LΛφ = −2ωφ − ciφ0, we obtain the following claim. Lemma 4.3. Let b ≥ 0 and c = 2κ0
√
ω. There exists C > 0 such that for s ∈ Iα1,
d ds(ε(s), iΛφω,c)L2 − (ε(s), 2ωφω,c+ ciφ 0 ω,c)L2 ≤ Ckε(s)k2H1. (4.9)
5. Proof of instability
We are now in a position to complete the proof of Theorem 1.3. We first note that by Lemma 3.5, the second term in the left-hand side of (4.9) is rewritten as
(ε(s), 2ωφ + ciφ0)L2 = ωMe(ε(s)) + c 2Pe(ε(s)) + O(kε(s)k 2 H1). By Lemma 3.6 we have ωMe(ε(s)) + c 2Pe(ε(s)) = ωMe(ε0) + c 2λ(s)Pe(ε0) = 2ω(ε0, φ)L2 + cλ(s)(ε0, iφ0)L2 + O(kε0k2H1).
Therefore, we obtain the following expression:
(ε(s), 2ωφ + ciφ0)L2 = 2ω(ε0, φ)L2+ cλ(s)(ε0, iφ0)L2
(5.1)
+ O(kε0k2H1) + O(kε(s)k2H1).
Proof of Theorem 1.3. Let α, β > 0 be chosen later and assume that ε0 ∈ H1(R)
satisfies
0 < kε0k2H1 ≤ β |(ε0, φ)L2| , ε0⊥ {χ, iφ, φ0, iφ0}.
Let u0:= φ + ε0 and let α satisfy
0 < α < min{α1, α2} < α0 < 1.
In what follows, we only consider the case (ε0, φ)L2 > 0 because one can treat the
case (ε0, φ)L2 < 0 in the same way.
Now suppose that u(t) ∈ Uα for all t ∈ R. Then it follows that Iα = R. From
Lemma 3.7, we have sup
s∈R
kε(s)k2H1 . α(ε0, φ)L2+ kε0k2H1.
We note that sups∈R|λ(s) − 1| . α by (3.15). Therefore, by Lemma 4.3 and (5.1), we have
d
ds(ε(s), iΛφ)L2 & (ε0, φ)L2− α(ε0, φ)L2 + O(kε0k
2 H1)
& (1 − α − Cβ)(ε0, φ)L2,
where the constant C is independent of ε0, α, β and s. Therefore, by taking α, β > 0
small enough, we obtain that d
ds(ε(s), iΛφ)L2 & (ε0, φ)L2 > 0 for all s ∈ R. This inequality yields that
(ε(s), iΛφ)L2 → ∞ as s → ∞.
On the other hand, from (3.15) we have sup
s∈R
|(ε(s), iΛφ)L2| . kΛφkL2 < ∞,
Appendix A. Relation to instability theory on (gKdV)
By following the argument of [8], we review the instability theory of the soliton Q(· − t) for the L2-critical generalized KdV equation
ut+ (uxx+ u5)x = 0, (t, x) ∈ R × R,
(gKdV)
and see a relation to our proof of Theorem 1.3. We define a tubular neighborhood around Q by
Uα= {u ∈ H1(R) : inf
y∈Rku − Q(· − y)kH
1 < α}.
The linearized operator L around Q is given by
Lv = −vxx+ v − 5Q4v for v ∈ H1(R).
We note that L satisfies the following properties:
LQ3= −8Q3, ker L = span{Q0}.
We consider the initial data u0 = Q + ε0 such that ε0 ∈ H1(R) satisfies
(ε0, Q3)L2 = (ε0, Q0)L2 = 0.
(A.1)
Let u(t) be the solution of (gKdV) with u(0) = u0. In the same way as in Section
3, one can prove that there exist α0 > 0 and C1-functions λ(t) > 0 and x(t) ∈ R
such that if u(t) ∈ Uα0 for all t ≥ 0, then ε(t) = ε(t, y) defined by
ε(t, y) = λ(t)1/2u(t, λ(t)y + x(t)) − Q(y) satisfies
(ε(t), Q3)L2 = (ε(t), Q0)L2 = 0 for all t ≥ 0.
(A.2)
We rescale the time t 7→ s by dsdt = λ31(t). A direct calculation shows that ε(s)
satisfies εs= (Lε)y+ λs λΛQ + xs λ − 1 Qy+ λs λΛε + xs λ − 1 εy − r(ε)y, (A.3)
where r(ε) is the sum of second and higher order terms of ε. By (A.2) and (A.3) one can prove that
λs λ + xs λ − 1 .kε(s)kL2 for all s ≥ 0. (A.4)
We now introduce the following functional J (s) = Z ε(s) Z y −∞ ΛQ, (A.5)
which corresponds to (4.7) as a Lyapunov functional. As pointed out in [8], if we consider the exponentially decaying data as
|ε0(x)| . ce−δ|x| for some δ > 0,
(A.6)
it is rather easy to show the L2-exponential decay on the right of the soliton. In particular, (A.5) is well-defined for all s ≥ 0. From (A.3) and (A.4), one can obtain easily that (A.7) d dsJ (s) = − Z ε(s)LΛQ − λs 2λ J (s) − 1 4 Z Q 2! + O(kε(s)k2L2).
Here we define a rescaled functional of J by K(s) = λ(s)1/2 J (s) − 1 4 Z Q 2! . It follows from (A.7) that
d
dsK(s) = −λ(s)
1/2
Z
ε(s)LΛQ + O(kε(s)k2L2),
which corresponds to (4.8). By using the relation LΛQ = −2Q, we have d
dsK(s) = 2λ(s)
1/2
Z
ε(s)Q + O(kε(s)k2L2),
which corresponds to (4.9). Therefore, if we assume (A.1), (A.6) and 0 < kε0k2H1 ≤ b0
Z ε0Q
(A.8)
for suitably small b0 > 0, we can complete the proof of instability of the soliton.
We conclude that the functionals (4.7) and (A.5) play an essential role in the proof of instability of the degenerate solitons in (1.1) and (gKdV), respectively, and that the unstable directions are determined by LΛφ for (1.1) and LΛQ for (gKdV), respectively.
Acknowledgments
N.F. was supported by JSPS KAKENHI Grant Number 20K14349 and M.H. by JSPS KAKENHI Grant Number JP19J01504.
References
[1] J. Angulo, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, Math. Surveys Monogr. 156, Amer. Math. Soc., Providence, 2009. [2] H. Bahouri, G. Perelman, Global well-posedness for the derivative nonlinear Schr¨odinger
equation, preprint [arXiv:2012.01923].
[3] T. Cazenave, Semilinear Schr¨odinger Equations, Courant Lecture Notes in Math. vol.10, Amer. Math. Soc., 2003.
[4] T. Cazenave, P.-L. Lions, Orbital stability of standing waves for some nonlinear Schr¨odinger equations, Comm. Math. Phys. 85 (1982), 549–561.
[5] H. H. Chen, Y. C. Lee, C. S. Liu, Integrability of nonlinear Hamiltonian systems by inverse scattering method, Phys. Scripta 20 (1979), 490–492.
[6] M. Colin, M. Ohta, Stability of solitary waves for derivative nonlinear Schr¨odinger equation, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 23 (2006), 753–764.
[7] A. Comech, D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math. 56 (2003), 1565–1607.
[8] L. G. Farah, J. Holmer, S. Roudenko, Instability of solitons-revisited, I: The critical generalized KdV equation, Nonlinear Dispersive Waves and Fluids, 65–88, Contemp. Math. vol. 725, Amer. Math. Soc., Providence, 2019.
[9] N. Fukaya, Instability of solitary waves for a generalized derivative nonlinear Schr¨odinger equation in a borderline case, Kodai Math. J. 40 (2017), 450–467.
[10] N. Fukaya, M. Hayashi, T. Inui, A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schr¨odinger equation, Anal. PDE 10 (2017), 1149–1167. [11] J. M. Gon¸calves Ribeiro, Instability of symmetric stationary states for some nonlinear
Schr¨odinger equations with an external magnetic field, Ann. Inst. H. Poincar´e, Phys. Th´eor. 54 (1991), 403–433.
[12] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), 160–197.
[13] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal. 94 (1990), 308–348.
[14] B. Guo, Y. Wu, Orbital stability of solitary waves for the nonlinear derivative Schr¨odinger equation, J. Differential Equations 123 (1995), 35–55.
[15] Z. Guo, C. Ning, Y. Wu, Instability of the solitary wave solutions for the generalized derivative nonlinear Schr¨odinger equation in the critical frequency case, Math. Res. Lett. 27 (2020), 339– 375.
[16] M. Hayashi, Potential well theory for the derivative nonlinear Schr¨odinger equation, to appear in Anal. PDE [arXiv:2011.08066].
[17] M. Hayashi, Stability of algebraic solitons for nonlinear Schr¨odinger equations of derivative type: variational approach, preprint [arXiv:2011.08029].
[18] M. Hayashi, T. Ozawa, Well-posedness for a generalized derivative nonlinear Schr¨odinger equation, J. Differential Equations 261 (2016), 5424–5445.
[19] N. Hayashi, T. Ozawa, Finite energy solutions of nonlinear Schr¨odinger equations of derivative type, SIAM J. Math. Anal. 25 (1994), 1488–1503.
[20] R. Jenkins, J. Liu, P. Perry, C. Sulem, Global existence for the derivative nonlinear Schr¨odinger equation with arbitrary spectral singularities, Anal. PDE 13 (2020), 1539–1578. [21] D. J. Kaup, A. C. Newell, An exact solution for a derivative nonlinear Schrodinger equation,
J. Math. Phys. 9 (1978), 789–801.
[22] M. Klaus, D. E. Pelinovsky, V. M. Rothos, Evans function for Lax operators with algebraically decaying potentials, J. Nonlinear Sci. 16 (2006), 1–44.
[23] S. Kwon, Y. Wu, Orbital stability of solitary waves for derivative nonlinear Schr¨odinger equa-tion, J. Anal. Math. 135 (2018), 473–486; Erratum: see arXiv:1603.03745, last revised on Oct. 30, 2019.
[24] S. Le Coz, Standing waves in nonlinear Schr¨odinger equations, Analytical and numerical aspects of partial differential equations, Walter de Gruyter, Berlin, 2009, pp. 151–192. [25] S. Le Coz, Y. Wu, Stability of multi-solitons for the derivative nonlinear Schr¨odinger equation,
Int. Math. Res. Not. IMRN (2018), 4120–4170.
[26] F. Linares, G. Ponce, G. Santos, On a class of solutions to the generalized derivative Schr¨odinger equations II, J. Differential Equations 267 (2019), 97–118.
[27] X. Liu, G. Simpson, C. Sulem, Stability of solitary waves for a generalized derivative nonlinear Schr¨odinger equation, J. Nonlinear Sci. 23 (2013), 557–583.
[28] M. Maeda, Stability of bound states of Hamiltonian PDEs in the degenerate cases, J. Funct. Anal. 263 (2012), 511–528.
[29] Y. Martel, F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal. 11 (2001), 74–123.
[30] Y. Martel, F. Merle, Blow up in finite time and dynamics of blow up solutions for the L2 -critical generalized KdV equation, J. Amer. Math. Soc. 15 (2002), 617–664.
[31] F. Merle, P. Raphael, The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schr¨odinger equation, Ann. of Math. 161 (2005), 157–222.
[32] K. Mio, T. Ogino, K. Minami, S. Takeda, Modified nonlinear Schr¨odinger equation for Alfv´en Waves propagating along magneticfield in cold plasma, J. Phys. Soc. 41 (1976), 265–271. [33] E. Mjølhus, On the modulational instability of hydromagnetic waves parallel to the magnetic
field, J. Plasma Phys. 16 (1976), 321–334.
[34] C. Ning, Instability of solitary wave solutions for the nonlinear Schr¨odinger equation of de-rivative type in degenerate case, Nonlinear Anal. 192 (2020), 111665, 23 pp.
[35] C. Ning, M. Ohta, Y. Wu, Instability of solitary wave solutions for derivative nonlinear Schr¨odinger equation in endpoint case, J. Differential Equations 262 (2017), 1671–1689. [36] T. Ogawa, Y. Tsutsumi, Blow-up of H1 solutions for the one-dimensional nonlinear
Schr¨odinger equation with critical power nonlinearity, Proc. Amer. Math. Soc. 111 (1991), 487–496.
[37] M. Ohta, Instability of bound states for abstract nonlinear Schr¨odinger equations, J. Funct. Anal. 261 (2011), 90–110.
[38] M. Ohta, Instability of solitary waves for nonlinear Schr¨odinger equations of derivative type, SUT J. Math. 50 (2014), 399–415.
[39] T. Ozawa, On the nonlinear Schr¨odinger equations of derivative type, Indiana Univ. Math. J. 45 (1996), 137–163.
[40] D. E. Pelinovsky, A. Saalmann, Y. Shimabukuro, The derivative NLS equation: global exis-tence with solitons, Dyn. Partial Differ. Equ. 14 (2017), 271–294.
[41] G. N. Santos, Existence and uniqueness of solution for a generalized nonlinear derivative Schr¨odinger equation, J. Differential Equations 259 (2015), 2030–2060.
[42] J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983), 313–327.
[43] J. Shatah, W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), 173–190.
[44] M. I. Weinstein, Nonlinear Schr¨odinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982), 567–576.
[45] Y. Wu, Global well-posedness for the nonlinear Schr¨odinger equation with derivative in energy space, Anal. PDE. 6 (2013), 1989–2002.
[46] Y. Wu, Global well-posedness on the derivative nonlinear Schr¨odinger equation, Anal. PDE. 8 (2015), 1101–1112.
[47] Y. Wu, Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension, preprint [arXiv:1705.04216].
(N. Fukaya) Department of Mathematics, Tokyo University of Science, Tokyo, 162-8601, Japan Email address: [email protected]
(M. Hayashi) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
